Noetherian ring: Difference between revisions

Revision as of 16:57, 17 January 2009

This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition
View a complete list of semi-basic definitions on this wiki

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Symbol-free definition

A commutative unital ring is termed Noetherian if it satisfies the following equivalent conditions:

Ascending chain condition on ideals: Any ascending chain of ideals stabilizes after a finite length
Every ideal is finitely generated

Definition with symbols

Fill this in later

Relation with other properties

Conjunction with other properties

Noetherian domain is a Noetherian ring that is also an integral domain.
Reduced Noetherian ring is a Noetherian ring that is also a reduced ring: it has no nonzero nilpotent elements.
Noetherian normal domain is a Noetherian ring that is also a normal domain.
Noetherian unique factorization domain
Local Noetherian ring is a Noetherian ring that is also a local ring.
Local Noetherian domain is a Noetherian ring that is also a local domain.

Stronger properties

Weaker properties

Coherent ring

Metaproperties

Closure under taking the polynomial ring

This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if
$R$
is a commutative unital ring satisfying the property, so is
$R [x]$

View other polynomial-closed properties of commutative unital rings

The polynomial ring over a Noetherian ring is again Noetherian. This is a general formulation of the Hilbert basis theorem, which asserts in particular that the polynomial ring over a field is Noetherian. Further information: Noetherianness is polynomial-closed

Closure under taking quotient rings

This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this property

View other quotient-closed properties of commutative unital rings

The quotient ring of a Noetherian ring by an ideal, is also Noetherian. Further information: Noetherianness is quotient-closed

Closure under taking subrings

This property of commutative unital rings is not closed under taking subrings; in other words, a subring of a commutative unital ring with this property need not have this property

A subring of a Noetherian ring is not necessarily Noetherian. For this, consider any non-Noetherian integral domain; this is a subring of a field, which is Noetherian.

Closure under taking localizations

This property of commutative unital rings is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.
View other localization-closed properties of commutative unital rings

A localization of a Noetherian ring is Noetherian. Intuitively, when we take localizations, we land up with fewer ideals, so the ascending chain condition becomes easier to satisfy. Further information: Noetherianness is localization-closed

Direct products

This property of commutative unital rings is finite direct product-closed: a finite direct product of rings with this property, also has this property
View other finite direct product-closed properties of commutative unital rings

A direct product of finitely many Noetherian rings is again Noetherian. This is essentially because ideals in the direct product look like direct products of ideals in the factors. For full proof, refer: Noetherianness is finite direct product-closed

Closure under taking completions

This property of commutative unital rings is completion-closed: the completion of a ring with this property, at any maximal ideal, also has this property
View other completion-closed properties of commutative unital rings

The completion of a Noetherian ring at a maximal ideal is again a Noetherian ring. For full proof, refer: Noetherianness is completion-closed

@@ Line 16: / Line 16: @@
 ==Relation with other properties==
+===Conjunction with other properties===
+* [[Weaker than::Noetherian domain]] is a Noetherian ring that is also an [[integral domain]].
+* [[Weaker than::Reduced Noetherian ring]] is a Noetherian ring that is also a [[reduced ring]]: it has no nonzero nilpotent elements.
+* [[Weaker than::Noetherian normal domain]] is a Noetherian ring that is also a [[normal domain]].
+* [[Weaker than::Noetherian unique factorization domain]]
+* [[Weaker than::Local Noetherian ring]] is a Noetherian ring that is also a [[local ring]].
+* [[Weaker than::Local Noetherian domain]] is a Noetherian ring that is also a [[local domain]].
 ===Stronger properties===
@@ Line 21: / Line 30: @@
 * [[Weaker than::Polynomial ring over a field]]
 * [[Weaker than::Artinian ring]]
-* [[Weaker than::Noetherian domain]]
 * [[Weaker than::Principal ideal ring]]
 * [[Weaker than::Dedekind domain]]
+* [[Weaker than::Cohen-Macaulay ring]]
+* [[Weaker than::Affine ring]]
 ===Weaker properties===